§19.37(iii) Legendre’s Incomplete Integrals

Functions F⁡(ϕ,k) and E⁡(ϕ,k)

Tabulated for ϕ=0⁢(5∘)⁢90∘, k2=0⁢(.01)⁢1 to 10D by
Fettis and Caslin (1964).

Tabulated for ϕ=0⁢(1∘)⁢90∘, k2=0⁢(.01)⁢1 to 7S by
Beli͡akov et al. (1962). (F⁡(ϕ,k) is presented as
Π⁡(ϕ,0,k).)

Tabulated for ϕ=0⁢(5∘)⁢90∘, k=0⁢(.01)⁢1 to 10D by
Fettis and Caslin (1964).

Tabulated for ϕ=0⁢(5∘)⁢90∘, arcsin⁡k=0⁢(1∘)⁢90∘ to 6D by Byrd and Friedman (1971), for ϕ=0⁢(5∘)⁢90∘,
arcsin⁡k=0⁢(2∘)⁢90∘ and 5∘⁢(10∘)⁢85∘ to 8D by
Abramowitz and Stegun (1964, Chapter 17), and for ϕ=0⁢(10∘)⁢90∘, arcsin⁡k=0⁢(5∘)⁢90∘ to 9D by
Zhang and Jin (1996, pp. 674–675).

Function Π⁡(ϕ,α2,k)

Tabulated (with different notation) for ϕ=0⁢(15∘)⁢90∘,
α2=0⁢(.1)⁢1, arcsin⁡k=0⁢(15∘)⁢90∘ to 5D by
Abramowitz and Stegun (1964, Chapter 17), and for
ϕ=0⁢(15∘)⁢90∘, α2=0⁢(.1)⁢1, arcsin⁡k=0⁢(15∘)⁢90∘ to 7D by Zhang and Jin (1996, pp. 676–677).

Tabulated for ϕ=5∘⁢(5∘)⁢80∘⁢(2.5∘)⁢90∘,
α2=-1⁢(.1)-0.1,0.1⁢(.1)⁢1, k2=0⁢(.05)⁢0.9⁢(.02)⁢1 to 10D by
Fettis and Caslin (1964) (and warns of inaccuracies in
Selfridge and Maxfield (1958) and Paxton and Rollin (1959)).

§19.37(iv) Symmetric Integrals

Functions RF⁡(x2,1,y2) and RG⁡(x2,1,y2)

Tabulated for x=0⁢(.1)⁢1, y=1⁢(.2)⁢6 to 3D by Nellis and Carlson (1966).

Function RF⁡(a2,b2,c2) with a⁢b⁢c=1

Tabulated for σ=0⁢(.05)⁢0.5⁢(.1)⁢1⁢(.2)⁢2⁢(.5)⁢5,
cos⁡(3⁢γ)=-1⁢(.2)⁢1 to 5D by Carlson (1961a). Here
σ2=23⁢((ln⁡a)2+(ln⁡b)2+(ln⁡c)2),
cos⁡(3⁢γ)=(4/σ3)⁢(ln⁡a)⁢(ln⁡b)⁢(ln⁡c), and
a,b,c are semiaxes of an ellipsoid with the same volume as the unit sphere.

Check Values

For check values of symmetric integrals with real or complex variables
to 14S see Carlson (1995).